This invention relates to apparatus and method for determining the manner by which an initial injection of signal energy is modified by a medium under test to a response state that is manifest as a redistribution of that energy.
There is for each event a defined scalar entity called total energy, E. The way in which this total energy is expressed in terms of the state variables, s, of an observation is called total energy density, E(s).
Energy density is a departure from equilibrium of those aspects of an event which are capable of doing work. It is composed of two parts: (1) that part representing the instantaneous configuration of work producing elements; and (2) that part representing the transformation that is in process in those work producing elements. The first part may be called potential energy density and designated V(s). The second part may be called kinetic energy density and designated T(s). Total energy density is thus given by the equation EQU E(s)=V(s)+T(s).
It is, and must be, a property of those partitions in energy density that their net sums are ultimately equal, i.e., that EQU .SIGMA..sub.s V(s)=.SIGMA..sub.s T(s)
Any manifestation of energy density, expressed in a frame of reference, will be proportional to the square of some observable expressed in that frame of reference.
There exists a unique relationship between V(s) and T(s) such that if EQU E= .sub.s E(s)ds&lt;.infin.,
and if there is some f(s) and g(s) such that EQU f.sup.2 (s)=V(s) EQU g.sup.2 (s)=T(s);
Then f(s) and g(s) are related by the Hilbert transform, and there is always a complex number EQU h(s)=f(s)+ig(s),
where i=.sqroot.-1
such that EQU .vertline.h.vertline..sup.2 (s)=.vertline.E(s).vertline..
Every linear observation is sufficiently described as a complex number h(s).
The expression of energy density and of its partitioning into potential and kinetic parts is what may be called the energy-coordinate expression, where the coordinate is one chosen. For example, if a frequency coordinate is chosen the term is energy-frequency, and if a time coordinate is used, the term becomes energy-time.
Although there can be many methods of expression, it is preferred that the energy-time, or energy-frequency, consist of two parts: (1) an amplitude, expressed in decibels or nepers, and (2) a phase, expressed in degrees or radians. Linear circuit theory already uses an expression coinciding with the energy-frequency expression. The power spectrum of a signal is proportional to the magnitude part, and the phase spectrum is proportional to the phase part. Thus, conventional frequency response is expressed as an amplitude in dB and a phase angle is energy-frequency.
The concept of energy-time, while mathematically straight forward, and physically meaningful, requires a realignment of conventional thinking. This is because we have been taught to think of frequency response as a complex measurement value, but time response as a scalar measurement value. But note that in a measurement in terms of just potential energy density, the value observed is indeed scalar. The imaginary part, kinetic energy density is nevertheless present and is treated in any mathematical treatment of the observed value as a Hilbert transform. For example, note that expressions involving energy are always present in mathematical processes involving the observed value, and more complete expressions involve both in-phase and quadrature components. Also note that cosine as well as sine terms are needed for general Fourier expansions. Thus, to bring scalar observations into alignment with the energy theorem, it is necessary that these characteristics of a Hilbert transform be present, except in calculations of simple time or frequency.
The inventor has disclosed in U.S. Pat. No. 3,466,652 a new measurement technique which can be used to break away from traditional time or frequency measurements. That technique produces what may be called a time delay spectrum. It is, in fact, a measurement technique which breaks away from the conventional time-frequency coordinates. That type measurement is in itself useful. However, one of the economically useful things which it can do for engineers and scientists is make conventional time-frequency measurements under physical conditions which would make more conventional techniques awkward. For example, making anechoic frequency measurement in an otherwise noisy and reverberant environment is virtually impossible with conventional techniques. Signals sent out from a single source of sound can reflect, refract and bounce many times before arriving at the measurement location. Thus the net sound measured is composed of a multiplicity of patterns arriving at different times and possessing different spectral properties. The classic frequency response from such a reverberation set is the steady state sinewave response in which all sound arrivals are combined into one grand spectrum. But often it is desirable to know what the spectrum would be for a particular sound following a particular path as though sound from the other signal paths were not present. The technique described in the aforesaid patent for such a particular path measurement is called time delay spectrometry (TDS).
The signal used in TDS has a constant total energy density and a uniquely defined partition into potential and kinetic energy densities. In the time domain, this signal takes the form EQU h(t)=e.sup.1.phi.(t)=cos .phi.( t)+i sin .phi.(t)
It is a property of this TDS signal that, when applied to a system under test, it illicits a response that is a mathematical hologram. However, unlike the more restricted class of optical hologram more familiar to all persons, the TDS hologram can alter dimensionality.
In an abstract sense, applying the TDS technique to a radiation signal of a system creates a response which is a hologram of the conventional response of that system. This holographic form is the function of the TDS processing for effectively slicing out of the system response only those things that are of interest. Those things can then be recorded in a form that has significance to the desired measurement. For example, applying the simple quadratic phase chirp EQU e.sup.i1/2ax.spsp.2
to a system will illicit a response that, in and of itself, has no significance in terms of the normal time domain response of that nework. In the case of a sound system, the response sounds like a complicated chirping of birds. To a listener, the sound is thus totally unintelligible because it is a hologram. The hologram has the property that is a two-dimensional representation involving the parameters of (1) time relative to the instant of stimulation, and (2) rate of change in phase. This representation can be made on a delay plane, i.e., two dimensional plane on which one dimension (coordinate) is time delay, and the other is the phase rate of the energy response of a transducer (e.g., microphone) at a point of interest in space remote from the signal source (e.g., speaker).
A tracking filter process can isolate all signals with a fixed time delay and produce a modified hologram that can be further processed for either (1) a smoothed frequency spectrum (energy density) of signals bearing a predetermined time delay between the source and transducer along a path of interest, which may be a direct path or a reflected path or (2) the time delay energy density of signals occupying a predetermined frequency band. It is because of this modality that the term "time delay spectrometry" is used.
TDS can use any constant total energy density phasoid EQU e.sup.i.phi.(x)
It is not limited to the quadratic phase chirp cited above for an example. However, certain circuit and/or algorithm simplicities result from the use of a quadratic chirp if the information desired is the impulse response or the Fourier transform of the impulse response. Consider a system that operates on a signal s(t) with a function f(t) to produce a response r(t), as follows: EQU r(t)=s(t) f(t)
where is the convolution operator. The system may be thought of as mapping the signal properties to the response space. The response is the image of the signal under the influence of the system.
An optical holograph is formed on photographic film as an interference pattern between a coherent illuminating wave used as a reference and the diffraction pattern of the object being photographed. In the same manner, an electronic hologram may be formed by the interference between a coherent signal from a source and the response of a system to that signal.
Converting an optical hologram to a scene in the original coordinate system requires coherent illumination of the hologram with the reference wave, and observation of stationary wavefront combinations. Similarly, stimulating an electronic hologram with coherent waves and summing the net behavior can convert the electronic hologram to the time coordinate called energy-time, or the frequency coordinate called energy-frequency.
Although reference will frequently be made hereafter to the sound field, it is ultimately the intent of every physical measurement to determine the manner by which an initial injection of signal energy is modified by a system under test to a response state that is manifest as a redistribution of that energy, either in the time or frequency domain, i.e., either in energy and time or energy and frequency coordinates discussed hereinbefore. In particular, the measurement of the energy of response as a function of the time coordinate is hereafter referred to as the energy-time measurement. A plot of this response will be referred to hereinafter as the energy-time curve (ETC). Energy is plotted as the ordinate, and time as the abscissa, in a Cartesian coordinate system, but that choice is one of convenience only. Any other two dimensional plotting system may be used. Consequently, the description of preferred embodiments which follow with reference to an ETC in the conventional Cartesian coordinate system is by way of example, and not limitation.
Within the framework of the foregoing general discussion of TDS as it will be applied to an ETC, it should be recognized that an ETC will consist of two parts, (1) the logarithmic magnitude as a function of time, expressed either in decibels or nepers, and phase angle as a function of time expressed either in radians or degrees of rotation. Thus, the entity from which these energy terms derive has the form of a complex signal representation having a real (or in phase) term and an imaginary (or quadrature) term. In classic linear theory, where there is total expression of the signal, this complex entity is called the analytic signal which takes the form EQU h(t)=f(t)+ig(t)
where i=.sqroot.-1, f(t) is the real part called impulse response, and i(g)t is the imaginary part related to the real part by the mathematical relationship known as the Hilbert transform.
Determination of the ETC is of value in establishing the time-delay properties of complicated systems, such as architectural acoustics where sound from a source will reach a point through many different paths, and therefore with many different time delays. TDS, as described in the aforesaid U.S. Pat. No. 3,466,652 makes possible in-place measurement of architectural acoustic response for any frequency band of interest which possesses a selected fixed time delay between the loudspeaker excitation and acoustic perception at a microphone as disclosed in that patent. The technique of TDS is applicable to many other complicated systems, and to other forms of radiation, such as electromagnetic radiation (radar or laser radiation, for example). Consequently, reference to wave radiation hereinafter should be construed to mean any wave radiation and not simply sound of a loudspeaker through the air, although that has been the primary field of application for TDS since at least Apt. 30, 1971 when the inventor presented a paper to the 40th Convention of the Audio Engineering Society, Los Angeles published in three parts in Journal of the Audio Engineering Society, (1971) Volume 19, Number 9, (Part 1, December, pp 734-743; Part II November, pp 829-834; Part III pp 902-905).
A technique by which ETC could be determined is described in Part I of that paper at pages 741 and 742. But such a technique, and other contemporary methods are extremely lengthy and complicated. An object of this invention is to provide frequency sweeping arrangements for TDS which invert the role of time delay and frequency: time delay appears in terms of frequency, and frequency components appear in terms of time delays. A further object of this invention is to provide a technique for quickly obtaining an ETC using a frequency-sweeping arrangement for TDS.